Hybridization of first and second sound in weakly interacting Bose gases

BEC Journal Club

July 16th

Lucas Verney

Summary

Sounds in superfluid 4He

Sound velocities in liquid Helium Sound modes in liquid Helium
Donnelly, Physics Today, 2009

  • Two sound velocities in a superfluid.
  • $u_2 < u_1$.
  • Pressure ($u_1$) / temperature ($u_2$) wave

What about BECs ?

Sound velocities


Pitaevskii & Stringari, BEC, OUP 2003

  • Two sound velocities
  • $u_2 < u_1$ ?
  • Hybridization at \[T \approx gn\]

Nature of the sound waves ?

  • Second sound is a wave where the normal and superfluid components oscillate with opposite phases.”, Hou, PRA, 2013

An interesting quantity is the Landau-Placzek ratio $\frac{c_p}{c_v} - 1$ which is always small in the case of He.

Taylor, PRA, 2009 and Landau, Fluid Mechanics


For BECs,

  • Below hybridization, Landau-Placzek ratio is small.
    Same behavior as in He:
    Pressure ($u_1$) / Temperature ($u_2$) wave.
  • Above hybridization, LP ratio is not small ⇒ very different behavior.

Landau's two fluid model

Description of the superfluid in terms of normal and superfluid component:

Sound modes in liquid Helium
Donnelly, Physics Today, 2009

\begin{equation} \left\lbrace \begin{array}{l} \vec{j} = \overbrace{\rho_s \vec{v}_s}^{\text{superfluid}} + \overbrace{\rho_n \vec{v}_n}^{\text{normal}}\\ \frac{\partial \rho}{\partial t} + \mathrm{div}(\vec{j}) = 0\\ \frac{\partial \vec{j}}{\partial t} + \vec{\nabla}{P} = 0\\ \\ \\ \frac{\partial \vec{v}_s}{\partial t} + \vec{\nabla}{\mu} = 0\\ \\ \frac{\partial (\rho \tilde{s})}{\partial t} + (\rho \tilde{s})\, \mathrm{div}(\vec{v}_n) = 0\\ \end{array} \right. \end{equation}

Linear equations ⇒ look for plane wave solutions.

Plane wave solutions must satisfy a fourth order equation on sound velocity $u$:

\begin{equation} u^4 - \left( \left.\frac{\partial P}{\partial \rho}\right|_{\tilde{s}} + \frac{\rho_s T \tilde{s}^2}{\rho_n \tilde{c}_v}\right) u^2 + \frac{\rho_s T \tilde{s}^2}{\rho_n \tilde{c}_v} \left.\frac{\partial P}{\partial \rho}\right|_{T} = 0 \end{equation}
  • Fourth order equation ⇒ 2 positive solutions.
  • Valid for any superfluid system: He + BEC.
  • Now we have to plug the right thermodynamic functions in this equation.
  • For BECs, we will use Bogoliubov thermodynamics at low T and Hartree-Fock to lowest order at high T.

Low temperatures

In this regime, we use Bogoliubov thermodynamics. The quasiparticles excitation spectrum is given by:

\[E(\vec{p}) = \sqrt{\frac{p^2}{2m}\left( \frac{p^2}{2m} + 2 g n\right)}\]

We express everything in terms of dimensionless units:

  • $\tilde{p} = \frac{p}{mgn}$
  • $\tilde{t} = \frac{k_B T}{gn}$

An Important parameter to have in mind is: \[\eta = \frac{gn}{k_B T_c} \approx 0.03\]

For example, for the free energy, we get:

\begin{equation} \frac{F(\tilde{t}\,)}{gn N}= E_0(na^3) + \eta^{3/2} \tilde{f}(\tilde{t}) \end{equation}

Express the normal density using Landau's formula:

\begin{equation} \rho_n = - \frac{1}{3} \int \frac{\mathrm{d} N_\mathbf{p}(\varepsilon)}{\mathrm{d} \varepsilon} p^2 \frac{\mathrm{d} \mathbf{p}}{(2\pi \hbar)^3} \end{equation}

Hybridization

  • Hybridization at \[k_BT = 0.6 gn.\]
  • Gap width $\propto \eta^{3/4}$.
  • If $\eta = 0$, crossing.

Below hybridization, same behaviour as in liquid helium:

  • First sound is an in-phase oscillation, a pressure wave.
  • Second sound is an out-of-phase oscillation, a temperature wave.

High temperatures

Ideal Bose gas thermodynamics except for \[\frac{1}{\chi_T} = \frac{gn}{m}.\]

For example:

\[\frac{\rho_n}{\rho} = \left(\frac{T}{T_c}\right)^{3/2}\]
\[\frac{S}{Nk_B} = \frac{5}{2} \frac{\zeta(5/2)}{\zeta(3/2)}\left(\frac{T}{T_c}\right)^{3/2}\]

  • First sound branches to the adiabatic “classical” sound velocity predicted by the ideal Bose gas model above $T_c$.
  • Second sound is mostly an isothermal oscillation of the superfluid component.

[WIP] A theory for all $T < T_c$ ?

Yes ! Beliaev theory. (Giorgini, New Journal of Physics, 2010).

Self-consistent equations. For example:

\[n = n_0 + \sum_{\mathbf{k}} \left[ \frac{\varepsilon_k + gn_0}{2E_k} (1+2N_E) - \frac{1}{2}\right]\]

where

\[\left\{\begin{array}{l} \varepsilon_k = \hbar^2 k^2 / (2m)\\ E_k = \sqrt{\varepsilon_k (\varepsilon_k + 2gn_0)} \\ N_E = \left(e^{\beta E} - 1\right)^{-1}\\ \end{array}\right.\]
  • Valid on a wide range of temperatures,
    (except for $T \approx 0$ and $T \approx T_c$).
  • Good agreement with Quantum Monte Carlo results.
  • Recovers Bogoliubov at low T, and Hartree-Fock at high T.

But:

Some problems with the derivatives ($c_v$, $\chi_T$).

Work in progress…

Conclusion

Thanks to S. Stringari, L. P. Pitaevskii, S. Giorgini.